Why my bone moves with the whole mesh
You have to reassign the influence of your eyelid bone to the eyelid vertices of your mesh.

As L0Lock said, you can fix your problem in Weight Paint mode, with the Mix or Substract brush.

You can also do it with the Vertex Groups "Assign" button:


Select your object, go in Edit mode, go in the Properties panel > Data > Vertex Groups.
In the vertex groups list, select the vertex group that has the name of your eyelid bone.
On the right of the vertex groups list, there's a down arrow that gives access to a menu. Press Remove from All Groups. Now there's no more vertices in this group, so the eyelid bone can't move anymore vertex.
In your mesh, select the vertices that are supposed to be part of the eyelid vertex group.
In the vertex groups, keep the eyelid vertex group select, and click on the Assign button that is under the list. You can modulate the assign ratio to give more or less influence to the brush.